461 



MATHEMATICS. ALGEBRA. 



[DIYMIOX. 



the operation is to proceed, like the oorrepondinz ope- 

 ration in arithmetic, till all the tenon of the dividend h.i vo 

 been brought down. Whon the list t -nn lias been brought 

 i, and annexed to the remainder no term in the 

 thvi.l .: i l,i\ ing been overlooked or neglected then the 

 row of term* thus got is the complete remainder. Should 

 it happen that the divisor will not go into this complete 

 remainder, the o[>eration is at an end, and you do just as 

 you would in the aimilar cao of arithmetic; you place 



this complete remainder, with the divisor underneath, in 

 the quotient, as a fractional correction. 



Divide ftr* ax o* by 2x o. Here dividend 

 and divisor are arranged according to the detcendittg 

 powers of x, the highest power of that letter occurring in 

 tho first term. If the arrangement be the reverse of this, 

 the operation will be that of the second form below ; 

 tniniu tho dividend is divided by minus the divisor for 



convenience. 11 



Cx 3ax 



Zax a 1 

 Sax a* 



2. o 2z)a + ax. 6x'(a 

 of 2ax 



Sax 6x> 

 Sax G* 1 



3. x + 2ajc+ ( 



10aV+ 10a'x s + 6ax + a s (x + 3^' + 3ax + a s 

 x* + 2ax -j- a V 



Ga'x 2 



+ 5aj; 



a'x 2 + 2ax + a 5 



4. x 2)2.e< 32 t(2x 3 + 4c'+8j; + 1C 

 2x 



32 

 16x 32 



0. <Lc 



+ l(3x + 2x 3 



21x 



^ 



14x' 



Bemainder, 



EXAMPLES FOR EXERCISE. 



2x + 5 



1. Diide ** ir 35 by x 7 



2. Diiidc * x 12 by * + 3 



3. Divide 6* + 13x 4- 6 by 3* + 2 



4. Divide 12* -f 29* 4 -4- U* by 3j' + 2x+ 

 Divide 18x 33x-' -f- 44x 35 by 3x" 



6. Diride x* y* by x y 



7. DUide x ox' 7ax + Po'x o 1 by x o 



8. Divide x + (a + 4) x + a* by x + o ' 



9. Divide 8x*y + 2x'y 2x 3xy 4 * by 4x'y-f3ry 1 



10. Divide x x -f x x' 1 by x' -f- x 1 



11. Divide 3 (2x a + 3) x 1 20x by 3 (x 1) * 



12. Divide x* -f px + f by x a 



13. Divide x* -j- jix' -f ?x + r by x a 



( It is worthy of notice, that in each of tho last two 

 examples, tho final remainder is the very same as tho 

 iividend, when the x in it is removed and a is written 



i: '. ,1 I 



Wbrthrr ont qaintiljr be dltldrd by mother, or mintu the furnifr by 



lwt- 



with mil tiplication : 



MM tb Utter, the result is the ssjne; for f is the tame as 

 trwcndi My stand for; It is of coarse Uu 



.jr tn or ; o ooarac te umo wt mi tipic 

 * 5S? fL?i ** ^ " P 1 "" H> '"I"" -,-. 

 Thi. >. nMraUr Uw MM M U + 0^ + Ox> +0-M; th 

 n M*d Mt IM bronchi down liU we trrire at th concJudin 



of U work. 



tt the 

 concJuding iten 



ON BXTONENTS, ROOTS, SURDS. 



We have- already stated (pnge 458) that there are two 

 ways of indicating a root of a quantity : one way by 

 means of tho radical sign /, placed before tho quantity, 

 :iud another by the introduction of a fraction, V.T: 

 in smaller character, over the right-hand corner of the 

 quantity proposed this corner fraction being called an 

 exponent or index. Tho radical sign is used exclusively 

 for roott: exponents are used alike for roots and for 

 powers thus, a* is the fourth power of a, and a} is the 

 fourth root of a. But exponents or indices have a wider 

 application still As yet we have had to deal only with 

 positive exponents ; algebraists, however, have introduced 

 negative exponents. We must give some account of 

 these showing how they have arisen, and what meaning 

 is attached to them. 



Your attention has already been drawn to tho fact 

 (page 458), that when a quantity with an exponent 



t In rxcrci*es in division, always expunge, before you brt>in thf 

 :<>n, whatever factor Is obviously common to both dividend niui 

 divisor, as you would do if you had to deal with :i fraction, the iii 

 being the numerator, and the divisor the denominator. Iti ih> i : tamplo 

 above, the factor x obviously enters all the terms of both dividend and 

 divisor; it is therefore a useless encumbrance, and should b>' i \punK< <l ; 

 the divlwm will then be 12* + 111 + 14 by It + 2. Example II admiu 

 of a like simplification. 



